fragmentation functions of pions and kaons in the nonlocal chiral quark model

Fragmentation functions of pions and kaons in the nonlocal chi-ral quark model Chung Wen Kao1,a, Dong Jing Yang2, Fu Jiun Jiang2, and Seng-il Nam3 1 Department of Chung Yuan Christian Un

Fragmentation functions of pions and kaons in the nonlocal chi-ral quark model

Chung Wen Kao1,a, Dong Jing Yang2, Fu Jiun Jiang2, and Seng-il Nam3

1 Department of Chung Yuan Christian University„ Chung-Li 32023, Taiwan

2 Department of Physics, National Taiwan Normal University, Taipei 11677, Taiwan

3 School of Physics, Korea Institute for Advanced Study (KIAS), Seoul 130-722, Korea

Abstract We investigate the unpolarized pion and kaon fragmentation functions using

the nonlocal chiral-quark model In this model the interactions between the quarks and

pseudoscalar mesons is manifested nonlocally In addition, the explicit flavor SU(3)

sym-metry breaking effect is taken into account in terms of the current quark masses The

results of our model are evaluated to higher Q2value Q2= 4 GeV2by the DGLAP

evo-lution Then we compare them with the empirical parametrizations We find that our

results are in relatively good agreement with the empirical parametrizations and the other

theoretical estimations

1 Introduction

The unpolarized fragmentation function Dh(z) represents the probability for a quark q to emit a hadron

hwith the light-cone momentum fraction z It can be written with the light-cone coordinate as follows :

Dhq(z, µ)= πz2

Z ∞

0

Dhq(z, k⊥, µ)dk2

⊥, Dh

q(z, k2T, µ) = 1

4z

Z

dk+Tr∆(k, p, µ)γ− |zk− =p − (1)

Here, k±=(k0± k3)/

√

2 and the correlation∆(k, p, µ) is defined as

∆(k, p, µ) =X

X

Z d4ξ (2π)4e+ik·ξh0|ψ(ξ)|h, Xihh, X|ψ(0)|0i, (2) where k, p indicate the four-momenta for the initial quark and fragmented hadron, respectively In ad-dition, z is the light-cone momentum fraction possessed by the hadron and µ denotes a renormalization scale at which the fragmentation process is computed Furthermore, k⊥is the transverse momentum

of the initial quark and kT = k − [(k · p)/|p|2] p is the transverse momentum of the initial quark with respect to the direction of the momentum of the produced hadron Finally X appearing above stands for intermediate quarks Notice all the calculations done here are carried out in the frame where the z-axis is chosen to be the direction of k Consequently one has k⊥ = 0 and kT , 0 in this frame Empirically, information of Dh(z) has to be extracted from the available high-energy lepton-scattering

a e-mail: cwkao@cycu.edu.tw

DOI: 10.1051/

C

Owned by the authors, published by EDP Sciences, 2014

, /201 06008 (2014)

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data by global analysis with appropriate parametrizations satisfying certain constraints For brevity,

we will simply refer to the unpolarized fragmentation functions as the fragmentation functions from now on

In Ref [1], the Nambu–Jona-Lasinio (NJL) model has been used to calculate the elementary fragmentation functions We have also employed the nonlocal chiral quark model (NLChQM) with the explicit flavor SU(3) symmetry breaking to calculate the elementary fragmentation functions [2, 3] These instanton-motivated approaches were used for computing the quark distribution amplitudes, manifesting the nonlocal quark-pseudoscalar (PS) meson interactions The elementary fragmentation functions are the functions in Eq (2) with the following approximation:

X

X

|h, Xihh, X| ≈ |h = q ¯Q, X = Qihh = q ¯Q, X = Q| (3)

Here h denotes the PS meson In other words, we just calculate the one-step fragmentation process: q(k) → h(p)+ Q(k − p) Here the PS meson h consists of the quark q and the anti-quark ¯Q In Refs [4–6], the NJL model has been applied for the fragmentation functions including the quark-jets and resonances The approach is actually applicable for any effective model Here we present the result of NLChQM including the quark-jet contribution

2 Inclusion of multi-jets contribution

To calculate the quark-jet contribution to the fragmentation functions within our model, we follow the approach in Refs [1, 4–6] The elementary fragmentation functions ˆdh(z) are re-defined as follows,

X

h

Z ˆ

dhq(z)=X

Q

Z ˆ

where the complementary fragmentation functions ˆdqQ(z) are given by

ˆ

dqQ(z)= ˆdh

The fragmentation functions Dhq(z) should satisfy the following integral equation:

Dhq(z)dz= ˆdh

q(z)dz+X

Q

Z 1

z

dy ˆdQ

q(y)DhQ z

y

! dz

Note that Dhq(z)dz in Eq (6) has a physical interpretation: Dhq(z)dz is the probability for a quark q

to emit a hadron which carries the light-cone momentum fraction from z to z+ dz ˆdQ

q(y)dy is the probability for a quark q to emit a hadron with flavor composition q ¯Qand a quark Q with the light-cone momentum fraction from y to y+ dy, at one stop Eq (6) actually describes a fragmentation cascade process of hadron emissions of a single quark

According to charge conjugation and isospin symmetry, there are only 11 independent elementary fragmentation functions Notice among these only four of them are not zero We call these direct fragmentation functions:

Dπu+(z) = Dπ −

d (z)= Dπ −

¯u (z)= Dπ +

¯

d (z), Dπu0(z)= Dπ 0

d (z)= Dπ 0

¯u(z)= Dπ 0

¯

d (z),

DKu+(z) = DK 0

d (z)= DK −

¯u (z)= DK 0

¯

d (z), DKs−(z)= DK 0

s (z)= DK +

¯s (z)= DK 0

¯s (z)

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0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

z

HKNS NJL nonlocal DSS

0.0 0.2 0.4 0.6 0.8 1.0

z

HKNS NJL nonlocal DSS

0.0

0.2

0.4

0.6

0.8

1.0

z

-HKNS NJL nonlocal DSS

0.00 0.05 0.10 0.15 0.20 0.25 0.30

z

HKNS NJL nonlocal DSS

Figure 1 The fragmentation functions zDπu0(z) (upper panel, right) and zDπu (z) (upper panel, left), zDK +

u (z) (bottom panel, right) and zDK−

s (z) (bottom panel, left) The uncertainty bands are according to HKNS parameter-izations

The other ones are called indirect fragmentation functions listed as follows:

Dπu−(z) = Dπ +

d (z)= Dπ +

¯u (z)= Dπ −

¯

d (z), DuK−(z)= DK 0

d (z)= DK +

¯u (z)= DK 0

¯

d (z),

DKu0(z) = DK +

d (z)= DK 0

¯u (z)= DK −

¯

d (z), DKu0(z)= DK −

d (z)= DK 0

¯u (z)= DK +

¯

d (z),

DKs+(z) = DK 0

s (z)= DK −

¯s (z)= DK 0

¯s (z), Dπs+(z)= Dπ −

s (z)= Dπ −

¯s (z)= DK +

¯s (z), Dπs0(z)= Dπ 0

¯s (z)

3 Results

We present our results at Q2 = 4 GeV2 and compare them with the empirical parametrizations and the NJL-jet model results We employ QCDNUM17 [7] to evolve our results from Q2 = 0.36 GeV2

to Q2 = 4 GeV2 Since Dπu+(z) is the most pronounced process, therefore, the initial momentum for evolution is determined by a reasonable agreement between our evolution result of Dπu+(z) with two empirical parameterizations, namely the HKNS parametrization [8] and the DSS parametrization [9] These two empirical parameterizations are used for comparison of other fragmentation functions as well Our result shows a good agreement with those parametrizations [10]

References

[1] T Ito, W Bentz, I -Ch Cloet, A W Thomas and K Yazaki, Phys Rev D 80, 074008 (2009) [2] S i Nam and C W Kao, Phys Rev D 85, 034023 (2012)

[3] S i Nam and C W Kao, Phys Rev D 85, 094023 (2012)

[4] H H Matevosyan, A W Thomas and W Bentz, Phys Rev D 83, 074003 (2011)

INPC 2013

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

z

-HKNS NJL nonlocal DSS

0.0 0.2 0.4 0.6 0.8 1.0

z

HKNS NJL nonlocal DSS

0.0 0.2 0.4 0.6 0.8 1.0

z

HKNS NJL nonlocal DSS

Figure 2 The fragmentation functions zDπu−(z) (left) and zDπs0(z)(middle) and zDπs (z)(right) The dashed lines denote the result of NJL model The uncertainty bands are according to the HKNS parameterizations

0.00

0.05

0.10

0.15

0.20

0.25

0.30

z

HKNS NJL nonlocal DSS

0.00 0.05 0.10 0.15 0.20 0.25 0.30

z

HKNS NJL nonlocal DSS

0.00

0.02

0.04

0.06

0.08

0.10

0.12

z

HKNS NJL nonlocal DSS

0.00 0.05 0.10 0.15 0.20 0.25 0.30

z

-HKNS NJL nonlocal DSS

Figure 3 The fragmentation functions zDK 0

u (upper panel, right) and zDK¯0

u (upper panel, left), zDK −

panel, right), and zDK +

s (bottom panel, left) The uncertainty bands are according to the HKNS parameterizations

[5] H H Matevosyan, A W Thomas and W Bentz, Phys Rev D 83, 114010 (2011)

[6] H H Matevosyan, W Bentz, I C Cloet and A W Thomas, Phys Rev D 85, 014021 (2012) [7] QCDNUM17, http://www.nikhef.nl/user/h24/qcdnum

[8] M Hirai, S Kumano, T H Nagai and K Sudoh, Phys Rev D 75, 094009 (2007)

[9] D de Florian, R Sassot and M Stratmann, Phys Rev D 75, 114010 (2007)

[10] D J Yang, F J Jiang, C W Kao and S i Nam, Phys Rev D 87, 094007 (2013)

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